On the Relationship between Radiance and Irradiance:
Determining the illumination from images of a convex Lambertian
object

Abstract

We present a theoretical analysis of the relationship between incoming
radiance and irradiance. Specifically, we address the question of
whether it is possible to compute the incident radiance from knowledge
of the irradiance at all surface orientations. This is a fundamental
question in computer vision and inverse radiative transfer. We show
that the irradiance can be viewed as a simple convolution of the
incident illumination, i.e.\ radiance and a clamped cosine
transfer function. Estimating the radiance can then be seen as a
deconvolution operation. We derive a simple closed-form formula for
the irradiance in terms of spherical-harmonic coefficients of the
incident illumination and demonstrate that the odd-order modes of the
lighting with order greater than one are completely
annihilated. Therefore, these components cannot be estimated from the
irradiance, contradicting a theorem due to Preisendorfer.

A practical realization of the radiance-from-irradiance problem is the
estimation of the lighting from images of a homogeneous convex curved
Lambertian surface of known geometry under distant illumination, since
a Lambertian object reflects light equally in all directions
proportional to the irradiance. We briefly discuss practical and
physical considerations, and describe a simple experimental test to
verify our theoretical results.

Summary

This paper derives an analytic formula for the irradiance in terms of
the radiance. The formula is in terms of spherical harmonics which
can be thought of in some sense as the eigenfunctions of the Lambertian
BRDF. It is shown that the odd-order modes of the illumination (greater than
one) are completely annihilated, so irradiance cannot be estimated from
radiance. There is also a deeper analysis of how positivity of the lighting
affects the results, and how this can allow for a solution in certain
special cases.

Finally, we present empirical and theoretical evidence showing that in
practice, the irradiance depends only on the first 2 orders of spherical
harmonic modes of the illumination, and can be represented as a quadratic
polynomial of the cartesian components of the surface normal. In fact,
99% of the energy of the Lambertian BRDF filter is contained by orders
0,1, and 2. In particular, we care only about the first 2 orders of the
illumination, that is 9 parameters. These first 9 coefficients are also
all that can be determined regarding the lighting. This 9 parameter model
is likely to have wide impact in computer graphics and vision, as we have
already demonstrated for forward and inverse rendering.

Results

The figures on the right summarize some of the main results from the
paper.

Figures 1,2:
Our analytic irradiance formula in terms of spherical harmonic coefficients,
and the analytic values for the coefficients of the Lambertian BRDF filter.
These are the key results of the paper. The coefficients of the Lambertian
BRDF fall off as l^(-5/2) for even terms and vanish for odd terms > 1. This
makes radiance-from-irradiance formally ill-posed as well as
ill-conditioned for frequencies l > 2.

Figure 3: A plot of the spherical harmonic coefficients (from
the equations in figure 2) versus frequency (l). We see the rapid decay of
coefficients and the zeros for odd modes greater than one.

Figure 4: Showing that we can indeed get the first 2 orders
(9 terms) of the lighting from the irradiance but that this fails miserably
for higher orders.

Relevant Links

JOSA 2001 paper in PDF (.4M) Correction: In equation 19, there is a small misprint.
The last term should be ((n/2)!)^2, not
(n!/2)^2